In this paper, we shall establish the superconvergence property of the Runge–Kutta discontinuous Galerkin (RKDG) method for solving a linear constant-coefficient hyperbolic equation. The RKDG method is made of the discontinuous Galerkin (DG) scheme with upwind-biased numerical fluxes coupled with the explicit Runge–Kutta algorithm of arbitrary orders and stages. Superconvergence results for the numerical flux, cell averages as well as the solution and derivative at some special points are shown, which are based on a systematical study of the \(\hbox {L}^2\)-norm stability for the RKDG method and the incomplete correction techniques for the well-defined reference functions at each time stage. The result demonstrates that the superconvergence property of the semi-discrete DG method is preserved, and the optimal order in time is provided under the smoothness assumption that is independent of the number of stages. As a byproduct of the above superconvergence study, the expected order of the post-processed solution is obtained when a special initial solution is used. Some numerical experiments are also given.

在本文中，我们将建立Runge-Kutta间断Galerkin（RKDG）方法的超收敛性，以求解线性常系数双曲方程。RKDG方法由不连续的Galerkin（DG）方案组成，该方案具有逆风偏向的数值通量，结合任意阶次和级的显式Runge-Kutta算法。基于对\（\ hbox {L} ^ 2 \）的系统研究，显示了数值通量，像元平均值以及在某些特殊点的解和导数的超收敛结果。-RKDG方法的标准稳定性和每个时间阶段定义明确的参考函数的不完全校正技术。结果表明，保留了半离散DG方法的超收敛性，并在与阶段数无关的平滑度假设下提供了最佳的时间顺序。作为上述超收敛研究的副产品，当使用特殊的初始溶液时，可获得后处理溶液的预期顺序。还给出了一些数值实验。